$\newcommand\ep\epsilon\newcommand\de\delta\newcommand\1{\mathbf1}\newcommand\R{\mathbb R}\newcommand\U{\mathscr U}$Previously, I suggested that for any probability distribution $\mu$ over $\R$ there exists a subset $A$ of $\R^n$ of nonzero Lebesgue measure such that each one-dimensional marginal of the uniform distribution $\U_A$ over $A$ is **exactly** $\mu$. I now doubt that. 

However, we can show the following: **For each natural $n$ and each probability distribution $\mu$ over $\R$** there is a family $(A^\ep\colon\ep>0)$ of subsets $A^\ep$ of $\R^n$ of nonzero Lebesgue measure such that each one-dimensional marginal of the uniform distribution $\U_{A^\ep}$ over $A^\ep$ converges (weakly) to $\mu$ (as $\ep\downarrow0$). 

Convolving $\mu$ with a distribution with a bounded continuous [mollifier][1] density (with respect to the Lebesgue measure over $\R$), we see that without loss of generality (wlog) $\mu$ has a bounded continuous density $p$. Let the desired set $A^\ep$ be of the form
\begin{equation*}
	A^\ep:=\{x\in\R^n\colon \|x-(x\cdot\1)\1/n\|_n\le\ep\, p(x\cdot\1/n)^{1/(n-1)}\}, \tag{1}
\end{equation*}
where $\cdot$ is the dot product over $\R^n$, $\1:=(1,\dots,1)\in\R^n$, and $\|\cdot\|_n$ is the Euclidean norm on $\R^n$. 

A more geometric description of the set $A^\ep$ is as follows. Consider  
the diagonal 
\begin{equation*}
	D:=\{t\1\colon t\in\R\}.  
\end{equation*}
Let $\Pi_t$ denote the affine hyperplane through the point $t\1$ perpendicular to the diagonal line $D$; let us refer to the point $t\1$ as the origin of the affine hyperplane $\Pi_t$. For each $x\in\R^n$ there is a unique real $t_x$ such that $x\in\Pi_{t_x}$; actually, $t_x=x\cdot\1/n$, the arithmetic mean of the coordinates of $x$. Then $A^\ep$ is the set of all points $x\in\R^n$ whose (shortest) (Euclidean) distance to the diagonal $D$ is $\le\ep\,p(t_x)^{1/(n-1)}$. 

Now take any real $a$. Let 
\begin{equation*}
	V_a:=\{(x_1,x_2,\dots,x_n)\in\R^n\colon x_1=a\}, 
\end{equation*}
the "vertical" $(n-1)$-dimensional affine hyperplane consisting of all points in $\R^n$ with "abscissa" $a$.
We want to show that 
\begin{equation*}
	\frac{|A^\ep_a|}{\ep^{n-1}}\to c p(a) \tag{2}
\end{equation*}
for some real $c>0$ (depending only on $n$), where $|A^\ep_a|$ is the Lebesgue measure of the "vertical" $a$-cross-section
$$A^\ep_a:=A^\ep\cap V_a$$
of $A^\ep$, consisting of all points in $A^\ep$ with "abscissa" $a$; that is, $|A^\ep_a|$ is the Lebesgue measure of the set $\{(v_2,\dots,v_n)\in \R^{n-1}\colon a\1+(0,v_2,\dots,v_n)\in A^\ep\}$. 

Since the density $p$ of $\mu$ is bounded, we have $p\le C^{n-1}$ for some real $C>0$. Since the diagonal $D$ does not lie on the affine hyperplane $V_a$, there is a real constant $b>0$ such that, if the distance of a point $x\in A^\ep_a$ from the point $a\1\in A^\ep_a$ is $>\ep Cb$, then the distance of $x$ from the diagonal $D$ is $>\ep C\ge\sup_s \ep p(s)^{1/(n-1)}$, which implies that $x\notin A^\ep$. (Here, one can take $b=\sqrt n$.) So, the distance of any point $x\in A^\ep_a$ from the point $a\1\in A^\ep_a$ is $\le\ep Cb$, and hence this point $x$ lies on the affine hyperplane $\Pi_t$ for some real $t$ such that $|t-a|\le\ep Cb$. Let now
\begin{equation*}
	m^\ep_a:=\min\{p(t)\colon |t-a|\le\ep Cb\},\quad M^\ep_a:=\max\{p(t)\colon |t-a|\le\ep Cb\}. 
\end{equation*}
Then, by the continuity of $p$, 
\begin{equation*}
	m^\ep_a\to p(a),\quad M^\ep_a\to p(a). \tag{3}
\end{equation*}

Let now $Px$ denote the orthogonal projection of a point $x\in\R^n$ onto the hyperplane $\Pi_0$. 

It follows that 

(i) if the projection $Px$ of a point $x\in V_a$ is at distance $\le m^\ep_a$ from the origin (of both $\R^n$ and the hyperplane $\Pi_0$), then $x\in A^\ep_a$; 

(ii) if the projection $Px$ of a point $x\in V_a$ is at distance $>M^\ep_a$ from the origin, then $x\notin A^\ep_a$. 

Thus, for some real constant $c>0$ depending only on $n$,
\begin{equation*}
c\,(\ep m^\ep_a)^{n-1}\le	|A^\ep_a|\le c\,(\ep M^\ep_a)^{n-1}. 
\end{equation*}
So, (2) follows in view of (3). 

That is, the density of the first of the $n$ one-dimensional marginals of the uniform distribution $\U_{A^\ep}$ over $A^\ep$ converges pointwise to the density $p$ of $\mu$. So, by [Scheffé's lemma][2], the first of the $n$ one-dimensional marginals of $\U_{A^\ep}$ over $A^\ep$ converges to $\mu$ in total variation. 
The same holds for each of the $n$ one-dimensional marginals of $\U_{A^\ep}$, since the sets $A^\ep$ are invariant with respect to any permutation of the coordinates. $\Box$


[1]: https://en.wikipedia.org/wiki/Mollifier#:~:text=In%20mathematics%2C%20mollifiers%20(also%20known,generalized)%20functions%2C%20via%20convolution.

[2]: https://en.wikipedia.org/wiki/Scheff%C3%A9%27s_lemma